Section 6.1 Parametric Curves
Subsection 6.1.1 Definition of Parametric Cuves
The major goal of this course is the extension of calculus to functions with multiple variable inputs and/or outputs. In the review sections in Subsection 3.1.1, I introduced linear transformations, which were linear functions \(\RR^n \rightarrow \RR^m\text{.}\) I would like to also investigate non-linear functions. Vector-valued function are the first step in that direction: they have a single-input but multi-variable non-linear output. The first and most important interpretation of vector-valued function is as parametric curves.
A parametric curve interprets a vector-valued function as a position in some euclidean space depending on time; as such, parametric curves are used to talk about motion through space. The calculus of parametric curves is a way to understand the physics of such motion, covering both linear and angular velocity and acceleration in a nice, holistic approach. When considering parametric curves, I like to imagine the movement of point-like objects through space under the influence of various forces. Projectiles with gravity and air friction is one imporant example; the motions of planets, moons and satellite under gravity is another.
Considering the motion of stellar objects around a large gravity source (such as planets, asteroids and comets around the sun), I will use the caluculus of parametric curves to derive Kepler's laws of planetary motion from the basic assumptions of Newtonian mechanics. Kepler's laws predate Newton, but they were simply observed, not derived. The fact that Newton's physics, with multivariable calculus, can recover these observations from first principles of motion and gravity is a major accomplishment of that theory. In order to cover Kepler's laws in full, these notes also include descriptions of conics as parametric curves.
Subsection 6.1.2 Parametric Curves
Definition 6.1.1.
A parametric curve in \(\RR^n\) is a continuous function \(\gamma :[a,b] \rightarrow \RR^n\text{,}\) that is, a continuous vector-valued function defined on an interval. As is convention, I will typically use the symbol \(\gamma\) for an arbitrary parametric curve.
I can identify a parametric curve with its image: that is, I think of a curve as a decription of the set of points in \(\RR^n\) which are the output of the curve. In this interpretation, the curve describes motion along this set of points: it starts at the point \(\gamma(a) \in \RR^n\) and moves along the curve, ending at \(\gamma(b) \in \RR^n\) when it reaches the end of its domain.
The continuity condition is important, since a parametric curve is a connected path. I could define vector-valued functions which jump around, but these don't really fit the notion of a curve they don't describe reasonable motion through space.
For visualizing parametric curves, it is conventional to graph only the output or image of the curve. There is never a \(t\) axis in any of these graphs; instead, the variable \(t\) is the parameter of movement along the curve. Let me start with some basic examples.
Example 6.1.2.
The curve \(\gamma(t) = (\cos t, \sin t)\text{,}\) for \(t \in [0, 2\pi]\) traces out a circle, as in Figure 6.1.3.
Notice that I defined this curve on the domain \([0, 2\pi]\text{.}\) If I extend this domain, the curve just starts to retrace over the circle. It's good to observe that parametric curves can self-intersect and trace over themselves many times.
Example 6.1.4.
The curve \(\gamma(t) = \left(\frac{1}{t}, t \right)\) on the domain \(t \in \left[\frac{1}{5},5 \right]\) traces part of the graph of \(f(x) = \frac{1}{x}\text{,}\) as in Figure 6.1.5.
Example 6.1.6.
The curve \(\gamma(t) = (\cos 2t, \sin t)\) on the domain \(t \in [0, 2\pi]\) oscilates faster in the \(x\) direction than in the \(y\) direction, as in Figure 6.1.7.
Example 6.1.8.
A spiral in \(\RR^2\) is a parametric curve of the form \(\gamma(t) = (f(t) \cos t, f(t) \sin t)\) where \(f(t)\) is a monotonic continuous function. It resembles the circle, but the radius is either increasing or decreasing as the curve traces around the circle. The curve \(\gamma(t) = (2e^{\frac{t}{4}} \cos t, 2e^{\frac{t}{4}} \sin t)\) is a logarithmic spiral, as in Figure 6.1.9. For the logarithmic spiral, the parameter \(t\) be any real number: the spiral will spin inwards and outwards without end.
Example 6.1.10.
The curve \(\gamma(t) = (t \cos t, t \sin t)\) is the archimedian spiral, as in Figure 6.1.11. For this spiral, I assume the domain \(t \in [0,\infty)\text{;}\) the shape starts at the origin and spins outward.
Example 6.1.12.
The curve \(\gamma(t) = (t\cos t, t\sin t,t)\) on \([0,20]\) is a spiral in \(\RR^3\text{,}\) as in Figure 6.1.13.
Subsection 6.1.3 Parametric Curves in Polar Coordinates
Once I am comfortable with changing coordinate systems, I can use any coordinate system I wish to describe parametric curves. For example, polar coordinates in \(\RR^2\) are well suited to describing any kind of object that is circular in some sense. Curves in polar coordinates are given as \(\gamma(t) = (r(t), \theta(t))\) instead of \(\gamma(t) = (x(t),y(t))\text{.}\) The circle in parametric coordinates is \(\gamma(t) = (1, t)\text{;}\) if I write the components individually, I could write \(\theta(t) = t\) and \(r(t) = 1\text{.}\) The logarithmic spiral has components \(\theta(t) = t\) and \(r(t) = 2e^{\frac{t}{4}}\text{.}\) The archimedian spiral has components \(\theta(t) = t\) and \(r(t) = t\text{.}\) In this course, I'm not going to spend much time on parametric curves in alternative coordinate systems, but it is good to know that such things are possible.